An alternating quantity (current i or voltage V) is one whose magnitude changes continuously with time between zero and a maximum value and whose direction reverses periodically. Some graphical representation for alternating quantities Equation of Alternating Quantities (i or V). When a coil is rotated rapidly in a strong magnetic field, magnetic flux linked with the coil changes. As a result an emf is induced in the coil and induced current flows through the circuit. These voltage and current are known as alternating voltage and current (1) Equation : Alternating current or voltage varying as sine function can be written as i = i 0 sint = i 0 sin 2t = i 0 sin t T 2 and t T V t V t V V 2 sin 2 sin sin 0 0 0 where i and V = Instantaneous values of current and voltage, i 0 and V 0 = Peak values of current and voltage = Angular frequency in rad/sec, = Frequency in Hz and T = time period (2) About cycle (i) The time taken to complete one cycle of variations is called the periodic time or time period. (ii) Alternating quantity is positive for half the cycle and negative for the rest half. Hence average value of alternating quantity (i or V) over a complete cycle is zero. (iii) Area under the positive half cycle is equal to area under negative cycle. (iv) The value of alternating quantity is zero or maximum 2times every second. The direction also changes 2times every second. (v) Generally sinusoidal waveform is used as alternating current/voltage. (vi) At 4 T t from the beginning, i or V reaches to their maximum value. t i or V + – Triangular + – t i or V Sinusoidal i or V + – + Rectangular t ac super imposed on dc i or V t Negative half cycle V0 or i0 Positive half cycle + – 0 2T T/2 T/4 i or V t or + – t i or V
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Some graphical representation for alternating quantities
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Alternating Current 1
An alternating quantity (current i or voltage V) is one whose magnitude changes continuously with time between zero and a maximum value and whose direction reverses periodically.
Some graphical representation for alternating quantities
Equation of Alternating Quantities (i or V).
When a coil is rotated rapidly in a strong magnetic field, magnetic flux linked with the coil
changes. As a result an emf is induced in the coil and induced current flows through the circuit.
These voltage and current are known as alternating voltage and current
(1) Equation : Alternating current or voltage varying as sine function can be written as
i = i0 sint = i0 sin 2 t = i0sin tT
2
and tT
VtVtVV
2
sin2sinsin 000
where i and V = Instantaneous values of current and voltage,
i0 and V0 = Peak values of current and voltage
= Angular frequency in rad/sec, = Frequency in Hz and T = time period
(2) About cycle
(i) The time taken to complete one cycle of variations is called the periodic time or time period.
(ii) Alternating quantity is positive for half the cycle and negative for the rest half. Hence average value of alternating quantity (i or V) over a complete cycle is zero.
(iii) Area under the positive half cycle is equal to area under negative cycle.
(iv) The value of alternating quantity is zero or maximum 2 times every second. The direction also changes 2 times every second.
(v) Generally sinusoidal waveform is used as alternating current/voltage.
(vi) At 4
Tt from the beginning, i or V reaches to their maximum value.
t
i or V
+
–
Triangular
+
–
t
i or V
Sinusoidal
i or V
+
–
+
Rectangular
t
ac super imposed on
dc
i or V
t
Negative half cycle
V0 or i0 Positive half cycle +
–
0
2
T
T/2 T/4
i or V
t or
+
– t
i or V
Alternating Current 2
Note : If instantaneous current i (or voltage V) becomes 1/n times of it's peak value in
time t then
nπ
Tt
1sin
2
1 second.
Important Values of Alternating Quantities.
(1) Peak value (i0 or V0)
The maximum value of alternating quantity (i or V) is defined as peak value or amplitude.
(2) Mean square value )or(22
i V
The average of square of instantaneous values in one cycle is called mean square value. It is
always positive for one complete cycle. e.g. 2
12
0
0
22 VdtV
TV
T
or 2
202 i
i
(3) Root mean square (r.m.s.) value
Root of mean of square of voltage or current in an ac circuit for one complete cycle is called
r.m.s. value. It is denoted by Vrms or irms
2
...... 0
0
0
2
222
21 i
dt
dti
in
iii
T
T
rms
= 0.707 i0 = 70.7% of i0
similarly %7.70707.02
00 V
VVrms of V0
(i) The r.m.s. value of alternating current is also called virtual value or effective value.
(ii) In general when values of voltage or current for alternating circuits are given, these are
r.m.s. value.
(iii) ac ammeter and voltmeter are always measure r.m.s. value. Values printed on ac
circuits are r.m.s. values.
(iv) In our houses ac is supplied at 220 V, which is the r.m.s. value of voltage. It's peak
value is .3112002 V
(v) r.m.s. value of ac is equal to that value of dc, which when passed through a resistance
for a given time will produce the same amount of heat as produced by the alternating current
when passed through the same resistance for same time.
Note : r.m.s. value of a complex current wave (e.g. i = a sin t + b cos t) is equal
to the square root of the sum of the squares of the r.m.s. values of it's individual
components i.e.
22
22
2
1
22ba
bairms .
Alternating Current 3
(4) Mean or Average value (iav or Vav)
The average of instantaneous values of current or voltage in one cycle is called it's mean
value. The average value of alternating quantity for one complete cycle is zero.
The average value of ac over half cycle (t = 0 to T/2)
%7.63637.02
00
2/
0
2/
0
i
i
dt
dti
iT
T
av
of i0, Similarly %7.63637.02
00 V
VVav
of V0.
Specific Examples
Currents Average value
(For complete
cycle)
Peak value r.m.s. value Angular
frequency
i = i0 sin t 0 i0
2
0i
i = i0 sin t cos t 0
2
0i 22
0i 2
i = i0 sin t + i0
cos t
0 02 i i0
(5) Peak to peak value
It is equal to the sum of the magnitudes of positive and negative peak values
Peak to peak value = V0 + V0 = 2V0 rmsrms VV 828.222
(6) Peak factor and form factor
The ratio of r.m.s. value of ac to it's average during half cycle is defined as form factor. The
ratio of peak value and r.m.s. value is called peak factor
Nature of
wave
form
Wave form r.m.s
.
valu
e
averag
e value
Form factor
valueAverage
valuer.m.s.fR
Peak factor
valuer.m.s.
valuePeakpR
Sinusoida
l
2
0i 0
2i
11.1
22
41.12
Half wave
rectified
2
0i
0i 57.12
2 +
i or V
2
+
+
–
i or V
2 0
Alternating Current 4
Full wave
rectified
2
0i
02i
22
2
Square or
Rectangul
ar
0i 0i 1 1
Phase.
Physical quantity which represents both the instantaneous value and direction of
alternating quantity at any instant is called it's phase. It's a dimensionless quantity and it's unit
is radian.
If an alternating quantity is expressed as )sin( 00 tXX then the argument of )sin( t
is called it's phase. Where t = instantaneous phase (changes with time) and 0 = initial phase
(constant w.r.t. time)
(1) Phase difference (Phase constant)
The difference between the phases of currents and voltage is called phase difference. If
alternating voltage and current are given by )sin( 10 tVV and )sin( 20 tii then phase
difference = 1 – 2 (relative to current) or 12 (relative to voltage)
Note : Phase difference, generally is given relative to current.
The quantity with higher phase is supposed to be leading and the other quantity is
taken to be lagging.
(2) Graphical representation
+
i or V
2
+
i or V
+
–
Voltage (V) = V0 sin t Current (i) = i0 sin ( t – )
Phase difference = 0 – (– ) = +
i.e. voltage is leading by an angle (+ ) w.r.t. current
V i
t
Voltage (V) = V0 sin t Current (i) = i0 sin ( t + )
Phase difference = 0 – (+ ) = –
i.e. voltage is leading by an angle (– ) w.r.t. current
i e
t
Alternating Current 5
(3) Time difference
If phase difference between alternating current and voltage is then time difference
between them is given as
2T.D.
T
(4) Phasor and phasor diagram
The study of ac circuits is much simplified if we treat alternating current and alternating
voltage as vectors with the angle between the vectors equals to the phase difference between
the current and voltage. The current and voltage are more appropriately called phasors. A
diagram representing alternating current and alternating voltage (of same frequency) as
vectors (phasors) with the phase angle between them is called a phasor diagram.
While drawing phasor diagram for a pure element (e.g. R, L or C) either of the current or
voltage can be plotted along X-axis.
But when phasor diagram for a combination of elements is drawn then quantity which
remains constant for the combination must be plotted along X-axis so we observe that
(a) In series circuits current has to be plotted along X-axis.
(b) In parallel circuits voltage has to be plotted along X-axis.
Specific Examples
Equation of V and i Phase difference Time difference T.D. Phasor diagram
V = V0 sin t
i = i0 sin t
0 0
or
V = V0 sin t
)2
sin(0
tii
2
4
T
or
V = V0 sin t
)2
sin(0
tii
2
4
T
or
V = V0 sin t
)3
sin(0
tii 3
6
T
or
V
i i V
V
i
/ 2
i
V
/ 2
i
V / 2
V
i / 3
/ 3
i
V
V / 2
i
Alternating Current 6
Measurement of Alternating Quantities.
Alternating current shows heating effect only, hence meters used for measuring ac are
based on heating effect and are called hot wire meters (Hot wire ammeter and hot wire
voltmeter)
ac measurement dc measurement
(1) All ac meters read r.m.s. value. (1) All dc meters read average value
(2) All ac meters are based on heating effect of
current.
(2) All dc meters are based on magnetic effect of
current
(3) Deflection in hot wire meters : 2rmsi
(non-linear scale)
(3) Deflection in dc meters : i
(Linear scale)
Note : ac meters can be used in measuring ac and dc both while dc meters cannot be
used in measuring ac because the average value of alternating current and voltage over a full cycle is zero.
Terms Related to ac Circuits.
(1) Resistance (R) : The opposition offered by a conductor to the flow of current through it is defined as the resistance of that conductor. Reciprocal of resistance is known as conductance
(G) i.e. R
G1
(2) Impedance (Z) : The opposition offered by the capacitor, inductor and conductor to the
flow of ac through it is defined as impedance. It’s unit is ohm(). rms
rms
i
V
i
VZ
0
0
(3) Reactance (X) : The opposition offered by inductor or capacitor or both to the flow of
ac through it is defined as reactance. It is of following two type –
(i) Offered by inductive circuit (i) Offered by capacitive circuit
(ii) LLX L 2 (ii)
CCXC
2
11
(iii) 0dc so for dc, XL = 0 (iii) For dc XC =
(iv) XL- Graph
(iv) XC - Graph
XL
XC
Alternating Current 7
Note : Resultant reactance of LC circuit is defined as X = XL ~ XC .
(4) Admittance (Y) : Reciprocal of impedance is known as admittance .1
ZY It’s unit is
mho.
(5) Susceptance (S) : the reciprocal of reactance is defined as susceptance .1
XS It is of
two type
(i) inductive susceptance LX
SL
L2
11 and (ii) Capacitive susceptance,
CCX
SC
C 21
.
Power and Power Factor.
The power is defined as the rate at which work is being done in the circuit.
In dc circuits power is given by P = Vi. But in ac circuits, since there is some phase angle between voltage and current, therefore power is defined as the product of voltage and that component of the current which is in phase with the voltage.
Thus cosiVP ; where V and i are r.m.s. value of voltage and current.
(1) Types of power
There are three terms used for power in an ac circuit
(i) Instantaneous power : Suppose in a circuit tVV sin0 and )sin(0 tii then
ViP ousinstantane )sin(sin00 ttiV
(ii) Average power (True power) : The average of instantaneous power in an ac circuit over a full cycle is called average power. It's unit is watt i.e.
cos2
1cos
2.
2cos 00
00 iViV
iVPPP rmsrmsavinstav 2
22
Z
RVRi rms
rms
(iii) Apparent or virtual power : The product of apparent voltage and apparent current in an electric circuit is called apparent power. This is always positive
2
00 iViVP rmsrmsapp
(2) Power factor : It may be defined as
(i) Cosine of the angle of lag or lead
(ii) The ratio Impedance
Resistance
Z
R
(iii) The ratio cos power Apparent
power True
kVA
kW
VA
W
Note : Power factor is a dimensionless quantity and it's value lies between 0 and 1.
For a pure resistive circuit R = Z p.f. = cos = 1
Alternating Current 8
Wattless Current.
In an ac circuit R = 0 cos = 0 so Pav = 0 i.e. in resistance less circuit the power consumed is zero. Such a circuit is called the wattless circuit and the current flowing is called the wattless current.
or
The component of current which does not contribute to the average power dissipation is called wattless current
(i) The average of wattless component over one cycle is zero
(ii) Amplitude of wattless current = i0 sin
and r.m.s. value of wattless current = sin2
sin 0iirms .
It is quadrature (90o) with voltage
Note : The component of ac which remains in phase with the alternating voltage is
defined as the effective current. The peak value of effective current is i0 cos and
it's r.m.s. value is cos2
cos 0iirms .
Concepts
If ac is produced by a generator having a large number of poles then it's frequency
2
second per rotation of polesNumber
2
nP
Where P is the number of poles; n is the rotational frequency of the coil.
Alternating current in electric wires, bulbs etc. flows 50 times in one direction and 50 times in the opposite direction in 1 second. Since in one cycle the current becomes zero twice, hence a bulb lights up 100 times and is
off 100 times in one second (50 cycles) but due to persistence of vision, it appears lighted continuously.
ac is more dangerous than dc.
The rate of change of ac is minimum at that instant when they are near their peak values.
ac equipments such as electric motors, are more durable and convenient compared to dc equipments.
Skin Effect A direct current flows uniformly throughout the cross-section of the conductor. An alternating current, on
the other hand, flows
mainly along the surface of the conductor. This effect is known as skin effect. the reason is
that when alternating current flows through a conductor, the flux changes in the inner part
of the conductor are higher. Therefore the inductance of the inner part is higher than that
of the outer part. Higher the frequency of alternating current, more is the skin effect.
The depth upto which ac current flows through a wire is called skin depth ().
Comparison of electricity in India and America India America
50 Hz 60 Hz
220 V 110 V
R R / 4 22
R
R
R VRP
VR (VR = rated voltage, PR = rated power)
i
i sin
i cos V
Iac = 0
Alternating Current 9
Example: 1 The equation of an alternating current is ti 400sin250 ampere then the frequency and
the root mean square of the current are respectively